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Rhythms related by small-integer-ratios (integers less than 3) are marked with crosses and labeled. B) Colors correspond to cyclic permutations of the pattern which has been inferred (i.e. the expectation template which maximizes pm), and transparency the entropy. This figure, and other ternary plots presented in this work, were made using the python-ternary package 47. A listener’s event-based expectations in PIPPET are modeled by positing that they expect events to occur as an inhomogenous point process, with probability λ(φ), referred to as an expectation template. This consists of a summation of Gaussian distributions, each centered around a mean phase at which an event is expected, and scaled according to expectation strengths and precisions.
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Further, Jacoby et al. 25 reported a slight tendency for reproduction of a rhythm’s cylic permutation where the final interval is the longest. One approach to address this in pPIPPET would be re-weighting priors derived from music corpora to reflect the asymmetries demonstrated empirically (see ‘optimal priors’ in 2). Trials involved five iterations of simulated tapping, each with 10 repetitions of the stimulus rhythm.
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The present simulations were limited to configuring pPIPPET using (real-world) notated musical rhythms containing both simple and complex integer-ratio rhythms. More detailed simulations of specific empirical studies will also enable exhaustive analysis of the interaction between expectation precision and template probability in pPIPPET. In the Results section we use pPIPPET to simulate the results of existing empirical datasets. In three experiments we examine whether the observed performance of participants can be related to specific differences in prior expectations, and their influence on expectancy-based entrainment, as characterized by pPIPPET. In Experiment 1 we use the model to simulate listeners from distinct musical cultures synchronizing to two-interval rhythms 5, i.e. repeating cycles of two sounds, defined by two inter-onset time intervals. In Experiment 2 we extend the scope of the simulations to periodic three-interval rhythms in a large rhythm space, which were explicitly notated (using discrete interval categories) by Western musicians 1.
